In this scenario, \(\sim 1830\)  light seeds (average number over 10 merger tree simulations) form at \(z>20\) , while the formation of medium-weight seeds is extended towards lower redshifts, as shown in the upper left and central panels of Figure REF . The heavy seeds are the rarest ones, \(< 1\%\) , forming only during a relatively short period of time, as a consequence of their tight birth environmental condition requirements (see Figure REF ). The number of medium-weight and heavy seeds is significantly reduced if a higher intensity of the LW flux is required (i.e. a higher critical LW flux threshold, \(J_{cr}=1000 \times 10^{-21}\rm erg/s/Hz/cm^2/sr\)  is assumed; see lower panels of Figure REF , model R1000).
In addition, [1]} also discuss the dependence of their results on the adopted seeding prescription, showing how the number and redshift distribution of the seeds vary with the adopted critical metallicity threshold (\(Z_{cr}\) ; see [1]} for details). Note that, we expect that only a small fraction (\(<15-20\%\) ) of all the seed BHs forming along the hierarchical assembly of the QSO eventually contribute to the growth of the final SMBH (e.g., [3]}, [1]}).
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